As we already observed, \(\mathbb {D}\)  has constant Gaussian curvature \(-1\) . Therefore Cheng's inequality [1]} applies and yields that for any \(x\in \mathbb {D}\)  and for any \(r>0\)  it holds
\(\lambda _1^D(B^{\mathbb {D}}_r(x))\le \lambda _1^D(B_r^{\mathbb {H}^2}(\bar{x}))\, ,\) 
